International School for Advanced Studies
(SISSA)
PhD Thesis:
Exploring Hund’s correlated metals:
charge instabilities and effect of
selective interactions
Candidate:
Maja Berovic
Supervisor: Massimo Capone
Trieste
October, 2018
1 Introduction 1
2 Strong electronic correlations 5
2.1 The band theory of solids . . . 5
2.2 Fermi liquid theory . . . 6
2.3 Mott insulators. . . 8
2.4 Single-orbital Hubbard model . . . 9
2.5 DMFT . . . 10
3 Multi-orbital models and Hund’s physics 15 3.1 d-orbitals in transition-metal oxides . . . 15
3.2 Multi-orbital Hubbard model . . . 17
3.2.1 Anisotropic Coulomb interaction . . . 20
3.3 Kanamori model . . . 21
3.3.1 Ising form of Kanamori model . . . 24
3.3.2 Ising vs. Kanamori model . . . 25
3.4 "Janus" effect of Hund’s coupling . . . 26
3.4.1 Energetics of the Mott gap . . . 28
3.4.2 Density-dependence of the electron-electron correlations . . . 31
3.4.3 Charge correlations in Hund’s metals and orbital decoupling . . . 33
3.4.4 Orbital selectivity . . . 36
4 The Gutzwiller variational method 41 4.1 The Gutzwiller approximation for multi-orbital systems . . . 42
4.1.1 The expectation values in infinite lattice coordination . . . 43
4.1.2 Variational problem . . . 45
4.2 Reformulation of the Gutzwiller approach . . . 46
4.2.1 Mixed state representation. . . 47
4.3 Variational energy of the multi-orbital model . . . 49
4.3.1 An explicit example: The single band Hubbard model. . . 50
4.4 Final remark . . . 52
5 Hund’s metals: A tale of two insulators 55 5.1 Model and atomic multiplets . . . 55
5.2 Mott transition at ¯n= 3 and ¯n = 2 . . . 58
5.3 Extended phase diagram in the U-J plane . . . 62
5.3.1 Two insulators . . . 62
5.3.2 The Hund’s metal as a bridge between the two insulators . . . 65
5.3.3 Charge fluctuations . . . 66
5.3.4 Region J/U > 3/4 . . . 67
5.4 The case ¯n= 3. Mott and Hund insulators coincides . . . 67
5.5 Phase diagrams for negative J . . . 69
5.6 Conclusions . . . 71
6 Orbital dependent interactions 73 6.1 Non-uniform density-density interactions for J = 0 . . . 74
6.1.1 Asymmetric interactions . . . 75
6.1.2 α = 1. . . 76
6.1.3 α = 0.7. . . 77
6.1.4 β = 0.8 . . . 79
6.1.5 Summary diagram . . . 80
6.2 Effect of the Hund’s coupling . . . 80
6.3 Conclusions . . . 81
7 Compressibility enhancement in multicomponent Hubbard models 83 7.1 Compressibility in strongly correlated models . . . 84
7.2 Compressibility enhancement in Hund’s metals: "Ising" Hamiltonian . . . 85
7.3 Gutzwiller approximation results . . . 88
7.3.1 Three-orbital Model . . . 88 7.3.2 Five-orbital model . . . 92 7.3.3 Discussion . . . 95 7.4 Conclusions . . . 97 8 Conclusions 99 A First appendix 103 A.1 Derivation of the multi-orbital Hubbard Hamiltonian . . . 103
B Second appendix 107 B.1 Contraction terms with two fermionic lines . . . 107
B.2 Contraction terms with four and more fermionic lines . . . 107
B.3 Derivation of the expectation values in the infinite lattice coordination . . . 108
C Third appendix 111 C.1 Charge fluctuations in non-interacting and fully interacting limit . . . 111
References 112
1
Introduction
One of the greatest scientific adventures of the last decades has been the search for an under-standing of high-temperature superconductivity. This fight is nowadays strongly entangled with the field of strongly correlated electron systems. This thesis finds its place in this framework and, despite we will not address directly the physics of superconductors, it is motivated by the properties of these materials, and in particular by the iron-based family.
Though it has been more than thirty years since the great discovery of high-temperature superconductors in copper-oxides [1–3], even nowadays we are lacking a generally recognized understanding of this phenomenon and a recipe to increase the critical temperature. The discovery of high-temperature superconductivity in iron-based compounds [4] about a decade ago stimulated the search for common features between these two different families, iron- and copper-based compounds, in order to get some general picture of the possible mechanism for superconducting pairing in these systems.
At first glance a similarity between the doping-temperature phase diagram of iron-based parent compounds and the celebrated one for the cuprates is evident, as reported in Fig.1.1. The striking point is a long range antiferromagnetic ordering arising in proximity of the su-perconducting (SC) state. At least in the perspective of this thesis, the main difference is that the magnetic state of the the cuprates is a Mott insulator, while the parent compounds of the iron-based superconductors are metallic despite the spin-density wave ordering. Moreover, it is believed that the superconductivity appears in the neighborhood of other competing phases, once these phases are suppressed. Typically doping or pressure drives the system out of this low-T regime of the parent compounds, suppressing the present magnetism, and leading the system to the superconductive phase [5]. Furthermore, both families consist of the layers com-posed by the transition-metal atom (Fe and Cu, respectively), with ligand atoms alternating in the neighboring layers.
Indeed, in the last years Fe-based superconductors (FeSC) were sintetized in different shapes, structure and composition. Nonetheless, they all share a couple of common robust characteristics: an atomic layers with Fe in square lattice, and a pnictogen (P/As) or a chalcogen (S/Se/Te)1 positioned above or below the center of each square of iron ions, forming this way the interlayers between the Fe planes, as shown in Fig.1.2(a). The formed structure has tetragonal symmetry,
1This is were the popular names for these compounds "pnictides" and "chalcogenides" come from.
Figure 1.1: Schematic phase diagrams of copper- and 122 family of iron-based high-temperature super-conductors [6]: both electron doping and hole doping suppress the magnetism of the parent compounds and induce superconductivity under the characteristic superconducting dome. The blue lines between the paramagnetic and pseuodogap phases (blue shading) in the copper-based case represent crossover transitions, black lines between paramagnetic and antiferromagnetic phases are well-defined transitions. The non Fermi liquid behavior (purple shade) is present for the same doping where one observes the superconducing phase, for larger values of temperature.
which undergoes a structural transition in some parts of the phase diagram [7].
However, the major difference between the two families of superconductors is evident already at the level of electronic structure. Though in both of the families the major role is played by transition-metal atom, with bands arising from 3d-orbitals crossing the Fermi level and dictating the physical properties of the system, the different number of electrons in the d-shell and the different crystal-field splitting lead to two different pictures. In the cuprates, only one band (arising from dx2−y2 orbital) crosses the Fermi level, leading to theoretical descriptions based on
the single-band Hubbard model or related models. On another hand, in FeSC the crystal-field splitting is typically quite small with respect to the other energy scales, meaning that in such case almost all five d-orbitals have to be taken into consideration while constructing the effective model. At the very least, a three-orbital modeling for the so-called t2gorbitals is necessary to reasonably account for the electronic structure of these compounds, but for some phenomena it is necessary to consider the whole five-fold manifold. In principle, this could provide a very basic argument for the metallic character of the undoped FeSC, since Mott localization is much harder to get in multi-orbital systems due to the higher number of kinetic energy channels. In the rest of the thesis we will discuss these arguments, showing that the picture is quite more complicated, rich and interesting. Nonetheless, correlation effects have been observed in all the various families of FeSC (in increasing order): 1111 pnictides (such as LaFeAsO), 122 (such as BaFe2As2), 111 (such as LiFeAs) and, at the more strongly correlated end [8], the 11 chalcogenides (FeSe, FeTe). We will not list the numerous experimental evidences, but let us show, as depicted in Fig. 1.2 (b), the DFT+DMFT results for the mass enhancement m∗/m as a measure of the correlations of the system of the iron 3d-orbitals in the paramagnetic state, together with the results obtained from optical spectroscopy experiments and (angle-resolved) photoemission spectroscopy experiments [9]. It is still under debate, however, whether this different degree of correlations arises due to the structural difference [9] or an increase of the interactions [10–12].
3
The importance of the multi-orbital nature of FeSC in determining the strength and the nature of correlations in FeSC is stressed in countless theoretical studies [11,13–16]. Accounting for the presence of more than one relevant orbital, one needs to take into consideration also the effect of the inter-orbital interactions which are richer than the pure Hubbard repulsion. In particular, one has to deal with the Hund’s exchange coupling, which favors high spin configurations. In the presence of the Hund’s coupling, the effective interaction that electron "feels" depends on the orbital character of d-orbitals and spin alignment (which are in accordance with the Hund’s rules in an isolated atoms) [16]. This energy scale is sizable in atomic Fe [17] and it leads to the remarkable effects which represent both the motivation and the topic of the present thesis. Intuitively, one expects that the Hund’s rule coupling enhances the electron-electron correlation effects, owing to the suppression of atomic configurations that do not maximize the local magnetic moment. We will see that the interplay between Mott localization and the Hund’s coupling is however more involved than this simple expectation.
Figure 1.2: (a) Structural motif of the FeSC. Inset: Top view of the FeX trilayer, where X=As,P,S,Se,Te. The triad (a, b, and c) demonstrates the three crystallographic directions. (b) The DFT+DMFT-calculated mass enhancement m∗/m of the iron 3d-orbitals in the paramagnetic state and the low-energy effective
mass enhancement obtained from optical spectroscopy experiments and (angle-resolved) photoemission spectroscopy experiments [9].
Many direct consequences of this physics have been identified. The main concept arising from these studies has been the ’Hund’s metal’, a metallic phase displaying strong correlations and resisting to the Mott localization up to very large values of the interaction. It is important to stress that this phenomenology has been found for integer fillings different from half-filling (for example two electrons in three orbitals), including the case of six electrons in five orbitals which is a characteristic of undoped iron-based superconductors. In this regime one finds a particular sensitivity of the observables to the strength of the Hund’s coupling J, an anomalous magnetic response [18] connected to a finite-temperature spin-freezing [19], an effective decoupling between the orbitals [14,17,20] and other anomalies that we will address in this thesis. In recent works [17,21], the crossover between a regular metal and the Hund’s metal has been connected to the Mott transition for the half-filled system [11, 12, 14, 22, 23], identifying a density-dependent crossover which seems to host the main anomalies we have listed above, plus an enhancement or even a divergence of the charge compressibility [21].
the so-called orbital selectivity, namely the realization of different physics for electrons with different orbital index. As we mentioned above, one of the effects of the Hund’s coupling is to favor an effective low-energy decoupling between the different orbitals, measured by vanishing inter-orbital charge correlations. Once the orbitals are decoupled, even small differences in the bandstructure parameters are strongly emphasized when correlations are increased. This gives rise to orbital-selective physics, which is realized by the combination of electronic-structure effects (which split the degeneracy) and a sizable Hund’s coupling. This can lead either to an orbital-selective Mott transition, where some orbitals are Mott localized and others are not [24], but also to a strong differentiation in the effective masses within the same metallic state [14,25], found in earlier studies [9,10,27].
In this thesis we tackle some important questions raised by the above mentioned studies and in general by the physics of iron-based superconductors and other materials with an important effect of the Hund’s coupling. We will focus on the origin of the interaction-resilient Hund’s metal and present a new picture of the stabilization of this phase, which connects the Hund’s metal with charge-disproportionation instabilities.
After introductory sections, where we will introduce the physics of strong correlations with a particular emphasis on the effect of multi-orbital character and the Hund’s coupling, we will present the theoretical approach which we used in our original investigations, the Gutzwiller approximation.
The new results of this thesis will be presented in Chapters5,6and7. In Chapter5we will revisit the phase diagram of a three-orbital Kanamori model by drawing its phase diagram in the full U-J plane, considering even regimes which are usually discarded. In this way we will show that the Hund’s metal state can be seen as a sort of superposition between two strongly correlated insulators: a high-spin Mott insulator and a disproportionated Hund insulator where the spatial charge distribution is not homogeneous. The mixed-valence nature of the metallic state can survive up to the large values of U and J if those are such to make the two insulating solutions degenerate (or nearly degenerate). The picture is confirmed by comparing it with the global half-filling case, where no Hund’s metal can be found, and the case where the Hund’s coupling is negative, which can be realized if a Jahn-Teller electron-phonon coupling exceeds the Coulomb exchange integral. Also in this case we find a correlation-resilient metal when disproportionated insulating solutions are degenerate with the Mott insulator and a more standard Mott transition when there is only one insulating solution which is simultaneously stabilized by U and J.
In Chapter 6we discuss the effect of non-uniform Coulomb interactions on the physics of strong correlations, a topic which is directly connected with the possible orbital selectivity and it is motivated by realistic calculations of interaction parameters, which are typically not symmetric.
In Chapter7we study the charge compressibility of multi-orbital models with full orbital-rotation invariance, comparing Ising and Kanamori interactions and different values of the orbital degeneracy. In all cases we find a phase separation instability for interactions slightly larger than the critical U for the Mott transition in the half-filled system. The phase separation boundary extends in doping and U in different ways, according to the number of orbitals and the interaction form. While the precise results seem to be strongly dependent on the "details" of the model, an enhancement of the compressibility appears as a generic feature, which can be connected with many aspects of the phase diagram of the iron-based superconductors and other materials.
2
Strong electronic correlations
2.1
The band theory of solids
Band theory was the first theory that had a huge success in explaining the electronic properties of solids (electronic conductivity, optical response etc.). It was able to predict, by simple means, whether a system was expected to be metallic or insulating [28]. Consequently, this theory was considered some sort of "standard model" of the solid state.
The band theory of solids is based indeed on a rather strong approximation, namely assuming that the electrons behave as non-interacting. This means that adding or extracting one electron from the system will not leave any consequences on any other electron, since they, electrons, do not ’feel’ the mutual presence. From a mathematical point of view, the band theory assumes that the Coulomb interaction between electrons can be approximated by an effective single-particle potential which adds to the interaction with the lattice. This approximation may appear rude because the energy scales associated with the Coulomb repulsion are quite large and they can not be neglected. Nevertheless, the band theory turned out to be valid in many solid systems since a large number of compounds have the screened Coulomb interaction. The rationale for this success is that the electronic properties are mainly controlled by the valence electrons whose interaction is screened by core electrons leading to a weaker effective interaction. The connection between the low-energy properties of interacting electrons and a sea of non-interacting electrons is the basis of the Landau theory of normal Fermi liquids, a phenomenological theory based on the idea that the excitations of a system of interacting fermions have the same nature of those of a non-interacting gas provided that some effective parameters are used.
Within the band theory the electronic spectrum is constituted by a number of energy bands separated by forbidden energy regions called gaps. The very existence of bands and gaps is a consequence of the periodic potential provided by the ionic lattice. Therefore one can construct the many-body state of the system by populating all the possible single-particle energy levels while respecting the Pauli principle. imposing that two electrons can not have the same quantum numbers. The last occupied energy level is usually called Fermi energy or Fermi level εF. The last populated band, commonly called valence band, can be partially or completely filled. When the valence band is partially filled, the system is metallic, and the electrons in this band give rise to the electrical conduction and to most electronic properties. Since there are free electronic
states just above the Fermi level, the electrons can be excited with no energetic cost. On the another hand, in order to have an insulator, the number of electrons must be such to completely fill the last, valence band, separated by a gap from the next, empty, band. In this latter case we have a situation where in order to add or excite one of the existing electrons would require an energy cost of the energy gap between the lower exactly filled valence band and upper empty conduction band. We say that the system has the gap at the Fermi level. Hence in this situation the insulator is inert to the applied external field, meaning that it can not cause the flow of the carriers.
Within this scheme, the only way to have a phase transition between a metal and an insulator would be to modify the electronic population of the valence band. This can be done by chemical doping or by electrostatic gating.
The above picture can easily be connected with the chemical character of the material: each atom in the solids contains an integer number of electrons. It is clear, due to the spin degeneracy, that having an even number of electrons per unit cell will mean having the full band hence an insulator. Within the band theory this is an necessary but not sufficient condition to have a gap at the Fermi level. Accordingly, systems with an odd number will end up having partially filled band driving the system to the metallic state.
2.2
Fermi liquid theory
A challenge to obtain the theoretical descriptions of interacting many-body systems gave rise to many different theoretical approaches. Capturing the effects of Coulomb interactions between many electrons in a lattice was greatly simplified by the Fermi liquid theory [29–31], an approximative approach introduced by Landau in 1956. As a phenomenological description of weakly interacting fermionic systems, the Landau theory of normal Fermi liquids provides description of many properties of metals, explaining why some properties of an interacting fermion system are very similar to those of the Fermi gas (i.e. non-interacting fermions), and why other properties differ. Moreover it is even able to describe some superconducting state, which was noticed in many metallic systems at low temperatures.
In fact, the basic assumption of this phenomenological theory is that the energy and low-temperature elementary excitation of a system of interacting fermions, so called quasiparticles, are in one-to-one correspondence with the excitations of a system of non-interacting particles. This can be realized starting from the single-particle excitations and "turning on" adiabatically the interaction without changing the character of the excitations. Therefore the quasiparticle states (thought not being a true state of the interacting Hamiltonian) can be labeled with the same quantum numbers (particle number, spin, and momentum) as the non-interacting fermionic states. This correspondence is valid only for weak excitations at low-energy, meaning that the excitation spectrum remains close to the Fermi surface. In general a quasiparticle excitation has a finite lifetime at low energy, as opposed to non-interacting particle.
At this point we can say that most of the low energy properties of a Fermi liquid can be interpreted as the ones of an ideal gas but rather with renormalized parameters. One of the main characteristics of the Fermi liquids is the quasiparticles effective mass m∗, which, comparing to the bare non-interacting electron mass m, has a larger value, owing to the presence of the interactions (between the quasiparticles). The bare band mass m of an electron is defined, within the band theory, as a result of the movement of an electron in a periodic potential, which makes its motion different with respect to the free-electron one. Accordingly one can see the effective mass m∗, within the Fermi liquid theory, as the reduction of the mobility of the electron as a consequence of the interactions between the particles. Hence, it comes naturally to take the two
2.2 Fermi liquid theory 7
masses, m∗/m, as the measure of the degree of correlation of a system under the consideration. Previously described phenomenological idea of Landau Fermi liquid theory was later de-veloped by Abrikosov and Kalatnikov [32], who gave a formal derivation using diagrammatic perturbative expansion of the interaction. Therefore, in order to give an insight in m∗/m ratio, let us consider the self-energyΣ(k, ω) that represents the contribution ("correction") to the single particle energy (or effective mass) due to interactions between the particle and the rest of the system. Let us then focusing on the zero-temperature behavior, up to order ω, where the self-energy is purely real and can be expanded as
Σ(k, ω) ' Σ0(k, 0)+ ω∂Σ 0(k, ω) ∂ω 0 , (2.1)
where we denoted the real part of the self-energy as Σ0(k, ω). Here we have neglected the imaginary part of the self-energy, that corresponds to the decay of quasiparticles, since we assumes that the decay rate of quasiparticles in much smaller than their energy. This is justified at energies and temperatures much lower than F. If we further expand the self-energy and the dispersion around the Fermi momentumkF, we obtain the expression for the effective mass
m m∗ = 1 − ∂Σ0∂k(k,0) k F 1 − ∂Σ0(kF,ω) ∂ω 0 . (2.2)
Since the self-energy does not depend on momentum in the mean-field approaches we will be dealing with in the present work, the numerator of Eq. (2.2) becomes 1 giving
m∗ m = 1 − ∂Σ0(k F, ω) ∂ω 0 !−1 = 1 Z, (2.3)
where Z represents the quasiparticle weight, which plays the role in renormalizing the quasi-particle energy. Eq. (2.3) shows that the effective mass is given simply by the inverse of the quasiparticle weight. Starting from the metallic phase with Z= 1, decrease of Z can be related to an increased effect of correlation, and consequently a metal-insulator transition will occur when Zvanishes or, following Eq. (2.3) the effective mass diverges. This means that at the moment of transition the electrons do not move anymore, they localize, hence one should expect the huge enhancement of the effective mass comparing to the non-interacting value. Consequently, the effective mass carries the main description of the correlation effects as long as the system remains in a Fermi liquid state. In another words, the quasiparticle picture of Fermi liquid theory can be taken as a correct description of the metallic states of the systems of interest, breaking down at the moment of transition to insulator phase.
On another side, the Landau theory that holds for the weakly interacting electrons fails to describe the system as soon as the interactions become sufficiently large to drive a phase transition to an insulator, or possibly to a novel non Fermi liquid metals characterized by anomalous properties. This scenario occurs in materials with open d- or f -electron shells, whose orbitals are localized and the bands are narrow. In these materials the effect of the Coulomb repulsion between electrons is very pronounced, meaning that the mean-field theories can not be applied anymore.
2.3
Mott insulators
Despite its huge success, the band theory of solids failed to describe the behavior of some materials with open d- or f -electron shells, such as transition-metal oxides. In spite of having an odd number of electrons per unit cell, already in the late 30’s of the twentieth century experiments [33] have shown unambiguously that these materials behave as insulators. Since this insulating behavior clearly contrasts with the band theory of solids, we are facing a new class of insulators, different from those predicted by the band theory, which are usually referred to as band insulators.
Mott and Peierls [34] suggested that this might be the result of the presence of strong Coulomb repulsion between electrons, which, in case of transition-metal oxides, are relevant and must be treated with equal footing with the kinetic energy of the electrons. Seemingly, this is the point where the mean-field treatment of the electron-electron interaction within the band theory fails, since this theory can hold just for the weakly interacting systems that can be represented with this independent particle picture. The reason why the effect of the screened Coulomb interaction can not be neglected is because d-orbitals of transition-metal oxides are very localized and the electronic bands are quite narrow due to the small overlap of the two adjacent d-orbitals.
The physical picture proposed by Mott can be described in terms of a tight-binding model where the electrons experience a screened Coulomb repulsion U when two electrons with opposite spin are on the same lattice site (corresponding to an ion in the crystal). As we shall see, this is precisely the famous Hubbard model. Let us consider a system with the same number of electrons as lattice sites, which corresponds to a half-filled band in the non-interacting limit. Within this simple framework we can not only understand the existence of Mott insulators, but also a new kind of transition connecting a metal to an insulator without changing the number of carriers or, equivalently, the Fermi level.
We can theoretically imagine to tune the interaction strength. At U = 0 the system obviously describes a metal because we have a half-filled band. When we increase the interaction, we have a competition between the kinetic energy, which favors delocalization of the electronic wave-function, and the interaction, which imposes constraints to the motion of the electrons. It is easy to be convinced that, when U becomes much larger than the kinetic energy, the ground state of the system is given by one electron localized on each lattice sites. Every hopping starting from a similar state necessarily creates a doubly occupied site, which is energetically unfavored by the Coulomb repulsion. In this way the electrons stay on their sites, ie. they localize and the material becomes an insulator, or more specifically - the Mott insulator [35].
From this description we can easily see that, in a Mott insulator, the behavior of each electron depends on the state of the others, which is exactly what the adjective correlated means. The correlations between electrons can not, obviously, be captured by the mean-field treatment of the band theory. For this reason the materials where the interactions are so strong to give rise to violations or even the breakdown of the band theory of solids are called strongly correlated systems.
We remind that Mott localization can happen only when the number of electrons is equal to the number of sites, for this particular case, or, for more generic picture when dealing with multi-orbital systems, at any commensurate filling when we have an integer number of electrons per site. Otherwise doping the Mott insulator with either holes or electrons the system becomes metallic; the correlations between the electrons still remain, but without possibility to drive the system up to the Mott insulating state in the absence of any spatial symmetry breaking.
It is important to stress that Mott insulators are usually characterized by a long-range ordering of the spin (and orbital for more general models) of the localized electrons. It is natural
2.4 Single-orbital Hubbard model 9
that some residual interaction emerges between the spins, leading to real-space ordering. For the single-band Hubbard model, one finds a Heisenberg interaction with antiferromagnetic coupling which gives rise to an antiferromagnetic ordering on bipartite lattice or in the absence of substantial frustration. Nevertheless, in this work we focus on paramagnetic solutions, where we inhibit any magnetic or orbital ordering to focus on the intrinsic correlation effects induced by the interactions. This solution is also representative of the finite-temperature behavior above the ordering temperature in actual materials.
The main reason for the interest in Mott insulators and Mott metal-insulator transitions is that many remarkable phenomena happen "close" to Mott transitions, i.e., with little changes of control parameters like doping, pressure, magnetic field or others. Among these phenomena the appearance of high-Tcsuperconductivity by doping a copper oxide is certainly one of the most fascinating and speculated one. In general terms, one can rationalize the proliferation of interesting phenomena in terms of the fragility of the strongly correlated metallic state in proximity of Mott localization.
2.4
Single-orbital Hubbard model
As we anticipated in the previous section, the basic ideas behind Mott insulators can be described in terms of a simple tight-binding model which includes a screened short-range Coulomb repulsion. Even if the valence bands of most correlated materials have d or f character, which would imply a multi-orbital description, we can picture the physics within a model where only one orbital is considered on every lattice site. This situation is also relevant for the high-temperature superconducting cuprates, where every copper atom has nine electrons in the d-shell: eight out of nine electrons will fill those levels which are substantially below (and do not affect the physics of the system), and the only band that crosses the Fermi level is the dx2−y2
with one electron per orbital.
For simplicity we only consider near-neighbor hopping on a square or cubic lattice experi-encing an on-site Coulomb repulsion U between two electrons on the same site (obviously with opposite spin in order to fulfill the Pauli principle). The Hamiltonian reads
ˆ H= −tX hi jiσ c†iσcjσ+ U X i ˆni↑ˆni↓, (2.4)
where c†iσ is the creation and cjσ is the annihilation operator on two different lattice sites i and j, respectively, hence the first term, together with the hopping amplitude t (an overlap of the two neighboring Wannier orbitals with dependence on the lattice constant a), represents the hoppings between the neighboring sites. The second term contains the local U Coulomb repulsion between the electrons, knowing that ˆniσ = c†iσciσis the number operator.
For U = 0 we have a standard tight-binding approximation for independent electrons which can be diagonalized in momentum space giving rise to a single band of width W = 2td, where d is dimentionality, and it is proportional to the hopping amplitude t. On the other hand, for t= 0 the model reduces to a collection of isolated sites, each hosting one localized electron. If we do not allow for any magnetic ordering, the spins of the electrons are disordered and we have a trivial paramagnetic insulator, where the electronic motion is impossible because of the absence of hopping processes. These two trivial limits, where the system is respectively a metal and an insulator, must be connected by some transition. If we consider the model at fixed density, it contains only two energy scales, therefore the whole physics must be controlled by the ratio U/W.
Despite the simplicity the model can not be solved exactly, except for the two limiting case of one and infinite spatial dimensions where the Dynamical mean-field theory becomes exact, thus the search for a characterization of the Mott transition is still a serious challenge.
The main reason for this is that the transition is expected to occur for intermediate U/W ∼ 1, where no perturbative approach is possible, and that the competition between the two terms is a very fierce one. The kinetic term tends to produce delocalized solutions, while the interaction terms wants to localize the electrons in real space. For these reasons, the study of the Hubbard model, especially in two dimensions, has been based on numerical approaches or approximate analytical methods.
As stressed before, one can take a quasiparticle spectral weight Z as a measure of metallicity, which is equal to 1 in metallic case. Increasing the value of U/W ratio Z starts to decrease. When the local interaction, ie. the Hubbard repulsion between the electrons U overcomes the energy scale of the kinetic energy which is defined by the bandwidth W, the quasiparticle spectral weight becomes suppressed, reaching the zero value at the transition.
In this thesis we do not attempt to review all the different approaches focusing on the various aspects of the Hubbard model, and we focus on a class of studies devoted to the Mott transition and focusing on the metallic side, where a strongly correlated metal emerges by increasing the interaction strength.
Mott insulator is characterized by the two bands, lower and upper Hubbard band, separated with the gap of the order U, that can be pictured as shown in the last panel of Fig.2.1. Upper Hubbard band includes those states where we have the double occupied site and represents the energy cost, in contrast to the lower Hubbard band. When the strength of the interaction overcomes the bandwidth, the gap opens and the metal-insulator transition occurs.
Let us try to describe schematically the Mott transition. Let us recall (2.4) and rewrite the Hamiltonian ˆH −µˆn it in a slightly different way, for the later convenience
ˆ H = −tX hi jiσ c†iσcjσ+ U 2 X i X σ ˆniσ− 1 2 ! 2 . (2.5)
This form comes from rescaling the chemical potential such that at half-filling µ= 0 and the particle-hole symmetry is evident. If we now consider a system at half-filling, the density of states at large U (U > W) can be represented [36] as two peaks centered around F±U2, so-called upper and lower Hubbard bands, both with bandwidth W and the gap between them. This picture describes in the appropriate way the Mott insulating state, and the distance U between them represents the energy cost for having double occupied state in the upper Hubbard band. Further, decreasing U means that the gap between these two peaks is decreasing (the overlap between the atomic orbitals starts to increase as well as the tendency of the electrons to delocalize) and finally vanishes at a critical value of the interaction strength Uc = W, signalizing the Mott insulator-metal transition in the single-orbital case [36,37]. This "atomic-limit" description will be discussed in more details in Sec.3.4.1.
2.5
DMFT
So far we have seen that the correlation driven Mott metal-insulator transition, as one of the most intriguing phenomena in condensed matter physics, has been widely examined in numerous studies, particularly the single-orbital model (2.4) at half-filling [36,38–41]. The Brinkman-Rice picture [42] of the transition (within the frameworks of Gutzwiller approach [43] - see Chapter4) gives a good description of the quasiparticle spectral weight in the metallic regime,
2.5 DMFT 11
with no possibility of reproducing the upper and lower Hubbard bands. The disappearance of the quasiparticle peak signalizes the metal-insulator transition. On the other side, Hubbard’s picture describes metal-insulator transition in terms of continuous splitting of metallic band into an upper and lower Hubbard band defining this way an insulating state. Apparently, this description does not give any information about the quasiparticle spectral weight. Dynamical mean-field theory represents the theoretical approach that was the first to unify the Mott and Brinkman-Rice picture and therefore describe the transition from both insulating and metallic side.
The initial point for constructing Dynamical mean-field theory (DMFT) [44–47] comes from the study of Metzner and Vollhardt [48] and shortly after Müller-Hartmann [49, 50], who realized that the diagrammatic perturbation expansions of expectation values (related to a Hamiltonian under the consideration) simplify in the limit of infinite lattice coordination number z. Keeping only the zeroth order term of such expansion gives a simplified theory that is exact in the limit z → ∞. An important point in this limit is that one has to scale properly the hopping parameter t → √t
z to get a non-trivial solution for the physical properties of the system [48]. Moreover, within the limit z → ∞ all non-local contributions to the self-energy vanish, recovering the momentum-independence of the self-energy Σ(k, ω) z→∞= Σ(ω). The self-energy, though momentum independent, retains the full many-body dynamics, and can thus describe genuine correlation effects.
Figure 2.1: In the limit z → ∞ the original Hubbard model reduces to a dynamical single-site problem, which may be viewed as an impurity atom embedded in a dynamical mean-field. Electrons may hop from this atom to the mean-field and back, whereas the on-site interact is as in the original model.
The second important step for the realization of this method lies in the following simplifica-tion suggested by Georges and Kotliar [51]: the lattice model, such as the single-orbital Hubbard one that we have previously introduced, is mapped onto a purely local impurity model, which is described with an atom embedded in a non-interacting bath [44,52,53], as represented in Fig.2.2. The bath represents the effective medium and it contains all the information about the intra-atomic interactions. It is coupled to the impurity atom through a hybridization function ∆(ω) via exchange of an electron. The local configuration on the chosen atom will essentially fluctuate between all the possible local configurations, giving an information about the quantum evolution of the atom. Moreover, the impurity energy becomes equal to the lattice self-energy. Having self-energy local gives the self-consistency condition and the crucial advantage of DMFT approach.
Starting with an arbitrary choice of the self-energy of the system, one solves the corre-sponding impurity problem in order to get a new self-energy. This must be repeated until the self-energy does not change anymore. From the converged self-energyΣ(ω) it is possible to
compute the full spectral function A(ω) of the impurity. Since the spectral function is related to the density of electron states, its evolution with respect to increasing interaction U can provide us an information about the emergence of the Mott transition at a specific critical value Uc. In Fig.2.2we show the schematic evolution of the density of states while increasing the interaction strength U, which summarizes up the results of many different studies from Ref. [44,54–56].
Figure 2.2: Schematic plots of the evolu-tion of the density of states with respect to increasing interaction. This scheme is made particularly for the case of half-filled Hubbard model in the paramagnetic case, in scope of comparing different theoretical approaches. The value of U is increasing from top to bottom. The first four plots with a finite spectral weight refer to the metallic phase. The very last plots refer to the insulating phase. The Hubbard peaks, as a precursor of the Hubbard bands, are visible already in the metallic phase.
At U = 0 the system is represented by the free-electron density of states, with a bare bandwidth W. As soon as we start increasing the interaction, we can observe the coherent quasiparticle peak, but slightly dif-ferent from the one expected from the Brinkman-Rice picture (dashed region in Fig.2.2), due to the appearance of the symmetric spectral weight broadening. These two broad peaks are considered to be a precursor of the lower and upper Hubbard bands previously introduced in the Hubbard picture [36,41]. They belong to the in-coherent part of the spectral function and represent the high-energy excitations caused by electron-electron in-teractions. At this point we can observe the dual nature of electrons, having both localized character at high energies and metallic behavior with the formation of itinerant quasiparticles at low energies. Increasing fur-ther U/W the two peaks tend to move furfur-ther apart, while the low-energy peak gets reduced. Close to the Mott transition (U . Uc), this three peak structure gets more pronounced. The quasiparticle peak becomes nar-rower, keeping the height at F unchanged. Suddenly it vanishes at a critical coupling Uc, signalizing the Mott transition, and leaving on another side Mott gap be-tween the lower and upper Hubbard band developed around F ± U/2.
The low-energy physics can be described, to some extent, in terms of Fermi liquid theory. Using the fact that, within DMFT, the self-energyΣ(ω) describes the local correlations as momentum-independent, we can conclude that the width of the peak that is proportional to the quasiparticle weight Z ≡ (1 − ∂Σ/∂ω)−1coincides with the effective mass enhancement, ie. m/m∗= Z−1 = 1 − ∂Σ/∂ω|ω=0. This means that for vanishing quasipar-ticle spectral weight the Mott transition occurs through a divergence of the effective mass associated with the localization of the electrons. Hence we can interpret the metal-insulator Mott transition as a delocalization-localization transition, demonstrating the wave-particle duality of electrons. The factor Z will be later on intro-duced within the Gutzwiller approximation described in Chapter4, which reproduces qualitatively in a correct way the DMFT picture of the coherent peak.
In this chapter we gave a very brief insight to Dynamical mean-field theory, focusing on the relevant aspects that will be important for understanding the results of the Gutzwiller
2.5 DMFT 13
approximation that we will mainly use within this work. As stressed, Gutzwiller approach is sufficient if considering the metallic side of the transition. However, the Gutzwiller approxi-mation picture can be compared, confirmed and eventually improved by DMFT study, which provides the exact properties not only of metallic but also insulating many-body state in infinite lattice coordination.
So far we have considered the single-orbital Hubbard model, but similar picture can be generalized for the multi-orbital one. However, DMFT must be taken as an approximation for real materials. Nevertheless, calculations showed that within DMFT a large amount of physical properties of among all transition-metal based materials can be analyzed and later compared with experiments [55,57, 58]. Information about the lattice structure appears in the DMFT equation only through the local density of states. Seemingly, more realistic (and less symmetric) density of states do account for the different competitions between interactions and hence different results in the various phenomena of this picture. Hence DMFT is a good starting point for analyzing this many-body problem.
Solving the impurity model in a self-consistent way, one gets the solution of the initial many-body problem. Yet, this approach, thought a simplification comparing to the original lattice model, still requires different numerical methods for its handling. Solutions of the general DMFT self-consistency equations require extensive numerical methods, in particular Quantum Monte Carlo techniques [44,59–62], the Numerical renormalization group [54,63–65], Exact diagonalization [66], and other techniques. All of these approaches use the fact that in the limit of infinite spatial dimensions z → ∞ the Hubbard model effectively reduces to a dynamical single-site problem with the self-consistency condition (Fig.2.1). In this sense, the only approximation in DMFT is the negligence of spatial fluctuations. However, it takes full account of quantum fluctuations, so that it becomes a good approximation in the case where the spatial fluctuations are not important, as for example in the systems with large coordination numbers.
3
Multi-orbital models and Hund’s physics
The single-band Hubbard model is the paradigmatic model to understand the physics of electron-electron correlations and the Mott transition in their simplest realization. However, in a vast majority of compounds the electronic structure can not be approximated with a single band crossing the Fermi level and a description in terms of the single-band model is questionable. This observation is not surprising because the typical example of strongly correlated materials are based on transition-metal atoms, in which the 3d-orbitals are partially occupied with different number of electrons according to the atomic species of interest. The five d-orbitals lead in principle to five bands in the solid (or even more if oxygen degrees of freedom are included and/or spatial symmetry breaking leads to a larger unit cell). While in some special situation, like the cuprates, the combination of a splitting of the bands and a particular filling leads to a single-band description, in the most general case, the electronic structure requires the explicit inclusion of a given number of orbitals.
The change from one to more orbitals turns out to be far from trivial, leading us to quote (with a slight abuse) Phil Anderson and his famous phrase "More is different". A number of recent studies shows indeed that Mott physics reveals surprising new phenomena when we take into account more than one orbitals and the consequent richer form of the electron-electron interaction.
As a matter of fact, the properties of multi-orbital systems are very rich and they depend crucially on different parameters, like the number of electrons and orbitals, the energy splitting between orbitals and the exchange coupling J. This can give rise to a variety of metallic, insulating, or bad metallic regimes, leading to the richness of phenomena that we can designate generally as Hund’s physics. In this chapter we review the main aspects of the modeling of multi-orbital materials and some very important results which represent the starting point of the original investigations reported in this thesis.
3.1
d-orbitals in transition-metal oxides
Transition-metal atoms are characterize by a partially filled 3d-orbitals. This level is five-fold degenerate due to angular momentum `= 2, which implies that the z-component can assume the values `z= −2, −1, 0, 1, 2. When the atom is included into a lattice, the neighboring atoms,
which can be other transition-metals or ligands (like oxygen in oxides) induce a crystal-field that lifts (partially) the degeneracy of these d-orbitals. Depending on the different coordination (thetrahedral, ochtahedral, square planar etc.), the crystal-field can have different effects, hence d-orbitals can be split in different way. This point is of the crucial importance because it allows us to understand which are the active orbitals that have to be taken into account when defining the model.
In some materials where the degeneracy is totally lifted, such as in the case of copper-oxides, it can happen (if the bands are narrower than the crystal-field splitting) that only one band with a precise orbital character crosses the Fermi level, meaning that the single-orbital Hubbard model is sufficiently good choice for a description of the low-energy physics. In this case just two control parameters, interaction strength U and the bandwidth W, are enough for understanding the important effects of the system.
We discuss now, as a notable example, the case of a cubic crystal-field splitting, generated by a configuration where the transition-metal atom is surrounded by ligand ions with negative charge at the same distance along the three coordinate axes, ˆx, ˆy, ˆz.
In Fig.3.1we show the typical structure of the atomic energy levels in this situation. The cubic crystal-field elevates the energy of dx2−y2 and d3z2−r2 orbitals, which are extending in the
direction towards the ligand ions, with respect to dxy, dyz and dxzorbitals, which are extending in directions between the surrounding ligand ions. Hence the five d-orbitals split into a two-fold degenerate subset, eg ≡
n
dx2−y2, d3z2−r2
o
, and a three-fold degenerate subset, t2g ≡ n
dxy, dyz, dxz o
, as demonstrated in Fig.3.1. Since this splitting is usually quite large, only one of the subsets, eg or t2g, crosses the Fermi energy. However, for example, in iron-based superconductors, where the ligand atoms are either pnictogen or chalcogen atoms, the separation between eg and t2g is small and all the five d-orbitals are relevant for the description of the low-energy physics. Therefore we can have situations where the effective low-energy manifold is composed by two, three or five orbitals. Moreover, there can be other perturbations that further lift the remaining degeneracies or modify this hierarchy.
Figure 3.1: Crystal-field split of d-orbitals of transition-metal ion in a cubic crystal-field with an octahe-dral environment.
3.2 Multi-orbital Hubbard model 17
One can conclude that when several correlated orbitals contribute to the conduction bands, more parameters come into play, since the number of electron-electron interaction terms in-creases due to inclusion of orbital degrees of freedom. We will show that this leads to a rich phenomenology.
3.2
Multi-orbital Hubbard model
In the previous chapter we have considered the single-orbital Hubbard model. However, as stressed before, in more realistic materials one actually has to deal with more than just one single-orbital. This mean that, when we expand the Coulomb interaction on a basis of localized Wannier orbitals (or even simply atomic orbitals), we find a number of independent integrals which in turn correspond to different couplings appearing in the second quantization electron-electron interaction term (see AppendixA). This holds even if we limit ourselves to on-site interactions. The most general form of the local Coulomb interaction is obtained in AppendixA
and is written as Umm0m00m000 = Z drdr’w∗m(r)w ∗ m0(r’) , U(r − r’)wm00(r’)wm000(r) . (3.1)
If we take m= m0= m00 = m000, we obtain a Coulomb integral which naturally generalizes the expression for the Hubbard U of the single-band model. This integral contains the product of two electron densities (squares of the Wannier orbital wave-function). If instead we take m= m0 and m00 = m000, with m , m00, performing a change of variable label m00 → m0, we define an independent integral associated with the overlap between the electron densities of two different Wannier orbitals. Assuming rotational invariance between the orbitals U and U0do not depend on the orbital indices and can be computed as
U = Z drdr’ |wm(r)|2U(r − r’) |wm(r’)|2 U0 = Z drdr’ |wm(r)|2U(r − r’) |wm0(r’)|2 . (3.2)
It is clear from Eq. (3.2) that U > U0 because the overlap between two different Wannier orbitals will always be smaller than the overlap of an orbital with itself. This reflects the natural expectation that the Coulomb interaction is stronger for electrons belonging to the same orbitals (therefore sharing the same portion of space) than for electrons in different orbitals, which avoid each other more effectively.
Eq. (3.1) defines also other Coulomb integrals, where m, m0, m00, m000 assume different values that one can account also for some other independent integral, which rather depends of the choice of relevant orbitals and symmetry of the system. But let us, for the sake of simplicity, consider just U and U0, with the possibility of having any number of the orbitals in the system (from five possible d-orbitals), and with no crystal splitting, in order to understand better the effect of inter-orbital Coulomb interaction U0. It is clear, though, that in more realistic treatment of the problem the Coulomb repulsion should be taken as orbital dependent, since, say, repulsion between the dxyand the dx2−y2 orbital is larger than the one between the dz2 and the dx2−y2 orbital.
Nevertheless, at this point it is enough to consider the upper assumption of having equal U0 between all different orbitals.
Assuming N orbitals, the interacting Hamiltonian gets the following form: ˆ Hint = U X i,m ˆnim↑ˆnim↓+ U 0 X i,m,m0 ˆnim↑ˆnim0↓+ U0 X i,m<m0σ ˆnimσˆnim0σ, (3.3)
where the first term represents the intra-orbital density-density interaction, while the second and third one stand for inter-orbital density-density interaction between anti-parallel and parallel spins, respectively. Considering the limit of isotropic U0= U, and adding the kinetic contribution and chemical potential shift to the interacting Hamiltonian (3.3), the total Hamiltonian becomes
ˆ H = −X hi ji X mσ ti jmc†imσcjmσ+ U 2 X i X mσ ˆnimσ− 1 2 ! 2 . (3.4)
where we have considered just the intra-orbital hopping terms between the same orbitals m (see AppendixA). Introducing 1/2 in the Hamiltonian (3.4) is just a convention for the chemical potential µ, in order to imply the particle-hole symmetry in half-filling case. The Hamiltonian considered here has a full S U(2N) rotational symmetry with respect to spin and orbital degrees of freedom. Indeed, the form of (3.4) shows us that the second term depends just on the total charge since the only relevant term that remains (apart of constant shift) is proportional to ˆn = Pmσ ˆnmσ. This indicates that any rotation in the local spin-orbital space leaves the interacting term invariant under these transformations hence independent of the choice of the local basis set.
One can anticipate the existence of the Mott transition for any integer filling of the system x = n/2N, where n = 1, ..., 2N − 1. Indeed, for very large U, the situation where n electrons are localized on each site represents the ground state. Instead, the excited state where a charge excitation is created is separated from the ground state by a gap of order U. This generalizes the half-filling situation of the single-band model, where n= 1 is the only configuration which can lead to a Mott transition.
Several different studies [67–71] using different techniques have recovered and confirmed this result. Particularly interesting for our case is the one obtained with the Gutzwiller approach (see Chapter4) in Ref. [67]. The authors demonstrated that the critical interaction strength Uc for which the metal-insulator transition occurs can be expressed as
Uc(N, n)= 1 (2N − n) n
p
n(2N − n+ 1) + p(n + 1)(2N − n)2|¯| (3.5) where ¯ is the average energy per site in the uncorrelated case
¯ =X σ Z D()d = X σ ¯σ, (3.6)
and D() represents the density of states. In our specific case we will assume the semi-circular density of states with half-bandwidth D= 1 and average energy per spin ¯σ ' −0.2122. Using Eq. (3.5) for this particular case, one can obtain the left panel of Fig.3.2.
Obviously, Uc is a function of N and n. Same results were recovered numerically, using Gutzwiller variational approach, while studying the behavior of quasiparticle spectral weight Z as a function of interaction strength U, with the transition into a Mott phase for U > Uc(n, N). Right panel of Fig.3.2essentially shows that if we analyze for instance case N = 3, the values of Uc coincide with the ones predicted with formula (3.5). Note that Uc(n= 1) = Uc(n= 5) as well as Uc(n= 2) = Uc(n= 4). This is a result of our choice of density of states which is symmetric in this specific case.
3.2 Multi-orbital Hubbard model 19
Figure 3.2: Left panel: Critical interaction Ucfor different number of orbitals N and different fillings
n = 1, 2, ..., 2N − 1, as a function of n. Right panel: Quasiparticle weight Z as a function of U/D, for three-orbital models at different fillings; results are obtained with Gutzwiller technique.
Let us now consider in particular the half-filling case. Recalling Eq. (3.5), imposing n = N, one obtains:
Uc(N)= 8(N + 1)|¯σ| (3.7)
which coincides with the result obtained analytically in Ref. [71] using the Slave-spin mean-field technique and performing a perturbative expansion1around the atomic limit. Ref. [71] points out
that the critical interaction strength is a growing function of the atomic ground state degeneracy, meaning that Uc increases together with N. Another point is that Uc is the largest for the half-filling case for any number of orbitals.
Figure 3.3: Quasiparticle weight Z as a function of interaction strength U/D (where D is a semi-circle with half-bandwidth), for the N-orbital Hubbard model at half-filling (with, from left to right: N = 1, 2, 3, 4, 5). The results are obtained with Gutzwiller variational approach and they show a perfect accordance with the Slave-spin mean-field [72]. Inset: Dependence of the critical interaction strength Ucon N.
Fig.3.3displays numerical results for the quasiparticle weight Z together with values of the critical interaction Uc (inset), for different number of orbitals N, and confirms the predictions of Eq. (3.7). These results are obtained both using Gutzwiller and Slave-spin [72] approach.
Result (3.7) can be compared with more reliable DMFT approach in the limit of infinite lattice coordination and large N [68]. In this limit we get the exact large N behavior of the critical interaction Uc(N)at half-filling, which is shown to be linear in N, ie.
Uc(N) = 8N|¯σ|, (at large N) (3.8)
It is evident that Eqs. (3.7) and (3.8) match in the limit of large N, meaning that the Gutzwiller approximation (and other slave-techniques) becomes more accurate with increasing N.
3.2.1
Anisotropic Coulomb interaction
When the Coulomb interaction is completely isotropic, i.e. U = U0, we find that the critical U c scales proportionally to the number of orbitals N. However the assumption that the Coulomb repulsion between electrons in the same orbital is the same as for electrons in different orbitals is not likely to be realized in solids. We thus consider here the first and simplest generalization, in which we analyze the case where U0
, U [73, 74]. Indeed, the two Coulomb integrals presented with Eq. (3.2) do differ in real materials. Let us parameterize the different values defining α according to U0 = αU and give an explicit example for the three-orbital case, though the following argument holds for any multi-orbital model. We present our results using the Gutzwiller approximation which agrees with previous slave-particle approaches [73,74].
Figure 3.4: Quasiparticle weight Z as a function of interaction strength U/D (where D is a semi-circle with half-bandwidth), for fixed value of U0/U = α. The results represent the three-orbital Hubbard model and they are obtained using the Gutzwiller variational technique, showing the perfect accordance with the RIBS results [74].
Fig.3.4shows the behavior of the quasiparticle weight Z as a function of the intra-orbital interaction U, for fixed values α = U0/U and at half-filling. U
c reaches a maximum U0 = U, where we obviously recover the result discussed above, namely UN=3
c . In the opposite limit U0 = 0, the orbitals are completely decoupled and the system is reduced to three single-band Hubbard models, and therefore the transition occurs at UN=1
c , which is much smaller than U N=3 c
3.3 Kanamori model 21
(see Fig.3.3, for n= N = 1). Interestingly the connection between the two obvious limits does not appear smooth and regular, but it is extremely abrupt. Indeed, even for α very close to 1 we find a significant reduction of Uc which rapidly approaches small values. Already for α ≈ 0.5 we recover Uc = UcN=1and the result remains unaltered all the way down to the decoupled value α = 0 [73,74].
We notice that the reduction of Uc is not associated with a smaller atomic gap (see also Sec.3.4.1), since∆at = U regardless the value of α, but it is related to a complete quenching of the orbital fluctuations associated with the lifting of the degeneracy of the system for U > U0, as shown in Table3.1. We will turn to this point in Chapter6.
n Degeneracy Energy 0, [6] 1 0 , [3U + 12U0] 1, [5] 6 0 , [2U+ 8U0] 2, [4] 12 U0, [U + 5U0] 3 U, [2U + 4U0] 3 8 3U0 12 U+ 2U0
Table 3.1: Eigenstates and eigenvalues of the Hamiltonian (3.3) in the atomic limit. The boxed numbers denote the ground state degeneracies for U > U0.
3.3
Kanamori model
Besides the two independent integrals introduced by Eq. (3.2), which are associated to charge-charge interactions for intra-orbital and inter-orbital Coulomb interaction, one can account for the other two independent integrals that result from Eq. (3.1) (see AppendixA). These integrals are defined as J= Z drdr’w∗m(r)w ∗ m0(r’)U(r − r’)wm(r’)wm0(r) , J0= Z drdr’w∗ m(r)w ∗ m(r’)U(r − r’)wm0(r)wm0(r’) , (3.9)
and they are related to the exchange integrals which are responsible for what we shall call Hund’s physics. In fact, F. Hund formulated a set of rules that describe the effect of exchange interactions in degenerate atomic shells. These rules specify the ground state configuration of outer-shell electrons in isolated transition-metal atoms [75]: singly-occupied configurations in each orbitals are the first to be created, aligning as much as spins as possible in order to reach the configuration with the total spin S maximized (rule of ‘maximum multiplicity’). Once given S , total angular momentum L should be maximized. When there are no single spin configurations left, the extra electrons are used to create doubly occupied configurations. Fig.3.5shows how the ground state configuration is built for the various atoms of the 3d-block of the periodic table, dictated by the Hund’s rules. These rules indeed arise simply from the exchange interaction associated to the couplings (3.9).
Eventually, recent works [17,20,76–78] have shown that the Hund’s exchange coupling plays an important role in tuning the correlations in correlated metals. The main investigation in this thesis will be shaped following this idea. Hence, in order to understand better the effect of these rules on a physical systems that we are interested in, let us consider a convenient choice of
Figure 3.5: Ground state electron configurations and Hund’s rules for 3d-elements
a set of orbitals, where we can utilize corresponding spatial symmetries, that can simplify further the problem. For this purpose let us assume, as a specific case that will be of main interest in this work, d-orbitals of a transition-metal ion in a crystal-field with an octahedral environment. As we have already emphasized, the five d-orbitals are typically split into a two-fold degenerate eg, and three-fold degenerate t2g orbitals (see Fig.3.1). Let us now consider the case where the t2gtriplet describes the low energy physics. Since wave-functions wm(r)can be chosen real, as it stands for d-orbitals, one gets that J = J0, which represents the further simplification and is convenient for the evaluation of the rotationally-invariant Kanamori Hamiltonian. All other integrals apart U, U0, and J are equal to zero due to the axial symmetry of d-orbitals. Since we want to understand what is the origin of the strong electron correlations in multi-orbital systems, it comes natural to try to understand the effect of each of these Coulomb integrals, (3.2) and (3.9).
The interacting Hamiltonian for the set of t2gorbitals is first proposed by Kanamori [79] for describing ferromagnetic metals, though later on it turned out to be convenient for describing multi-orbital systems in general. It has the following form:
ˆ Hint= X m Uˆnm↑ˆnm↓+ X m,m0 U0ˆnm↑ˆnm0↓+ X m<m0σ U0− J ˆnmσˆnm0σ − J X m,m0 c†m↑cm↓c † m0↓cm0↑+ J X m,m0 c†m↑c†m↓cm0↓cm0↑, (3.10)
with number operator ˆnmσ = Pmσc†mσcmσ that counts electrons on orbital m = 1, 2, 3 with spin σ. The first three terms of the Kanamori Hamiltonian (3.10) expresses density-density interactions: the intra-orbital Coulomb repulsion U and the inter-orbital one U0between two electrons with opposite spins, as well as the inter-orbital Coulomb interaction U0− J between two electrons with aligned spins, respectively. The last one, U0− J, includes the z-component
3.3 Kanamori model 23
of Hund’s coupling J, and reflects Hund’s first rule. If we present these density-density terms schematically, as in Fig.3.6, it becomes obvious that the Coulomb repulsion U between two electrons occupying, say, orbital m (a) can be reduced placing one of the two electrons on a different orbital m0 (b), since U0 < U. Moreover, if the two electrons occupy two different orbitals, m and m0, the Pauli principle does not prevent the electrons to have parallel spins along the quantization ˆz-axis (c). In such configuration the energy is further lowered by the Hund’s coupling J, ie. it is equal to U0− J.
Figure 3.6: Electron-electron Coulomb interactions in multi-orbital systems: (a) intra-orbital, (b) inter-orbital with anti-parallel and (c) inter-inter-orbital with parallel spins
The very last terms in Eq. (3.10) represent the spin-flip and pair-hopping interactions: the former is the x- and y-component of Hund’s exchange and it flips the spins of two singly-occupied orbitals (Fig. 3.7(a)), while the latter represents the two-electron transfers from a doubly-occupied to an empty orbital (Fig. 3.7 (b)). These two terms are needed in order to preserve the S U(2) spin symmetry of Kanamori Hamiltonian, which is still not evident from the form of Eq. (3.10).
Figure 3.7: Hund’s exchange processes: (a) spin-flip and (b) pair-exchange
Furthermore, in order to have the rotational invariance of the system, one needs to impose an additional constraint, ie.
U0 = U − 2J . (3.11)
This condition is exact in case of equivalent t2gorbitals which actually are invariant under the rotations in real space. Obviously, this holds just when the crystal splitting is strong enough to push away egpair of orbitals, and if U, U0and J are calculated assuming a spherically symmetric interaction, as we have done so far.
The choice of t2gorbitals is relevant for describing the transition-metal oxides with cubic symmetry. In the solid-state, however, in many different materials where we consider orbitals different that t2gones, the spherical symmetry of the screened Coulomb interaction U(r − r0) is not preserved, hence the Kanamori Hamiltonian (3.10) is not exact. Nevertheless, numerous studies applied the Kanamori interaction (3.10) to the full set of d-states. This choice is often considered to be a reasonable approximation so that U0 = U − 2J can be used in order to ease the numerical calculations and simplify the definition of the problem. However, for more
realistic treatment of screened interaction one should consider the full Coulomb interaction, which contains also different matrix elements, and/or realistic values of the parameters of the Hamiltonian, which can be obtained through constrained-RPA (Random-phase approximation) method [80].
The full rotational symmetry (ie. invariance under charge, spin and orbital gauge trans-formations, separately, denoted as U(1)C ⊗ S U(2)S ⊗ S O(3)Osymmetry) becomes apparent if we rewrite the Kanamori Hamiltonian (3.10) in a slightly different way. This can be done by defining the local electron number, spin and orbital angular momentum operators, respectively:
ˆn= X mσ c†mσcmσ, S= 1 2 X m, σσ0 c†mσσˆσσ0cmσ0, L= X mm0, σ c†mσ`ˆmm0cm0σ, (3.12)
where m, m0 = 1, 2, 3 labels the t
2gorbitals and σ=↑, ↓ the spin components, whereas ˆσσσ0 are
the Pauli matrices and ˆ`(m)m0m00 = −imm0m00 are the O(3) group generators characterizing orbital
rotations. Once done so, the Hamiltonian (3.10) becomes [78]
ˆ Hint = (U − 3J) ˆn(ˆn − 1) 2 − J " 2 S2+ 1 2L 2 # + 5 2Jˆn . (3.13)
The first term of Eq. (3.13), that depends on the total charge of the site and is proportional to the Hubbard U, can be interpreted as the overall Coulomb repulsion experienced by electrons on the same site. On another hand, the second term represents the Coulomb exchange mechanism responsible for Hund’s rules, which favors, in following order, high spin and high orbital angular momentum configurations. Therefore, J tends to maximize S and L obeying the Hund’s rules, in order to minimize the energy of the system, according to Eq. (3.13).
The spectrum of Hamiltonian (3.13) is given in Table3.2:
n ` s Degeneracy Energy 0, [6] 0 0 1 0 , [15U − 30J] 1, [5] 1 1/2 6 0 , [10U − 20J] 1 1 9 U −3J , [6U − 13J] 2, [4] 2 0 5 U − J, [6U − 11J] 0 0 1 U+ 2J , [6U − 8J] 0 3/2 4 3U − 9J 3 2 1/2 10 3U − 6J 1 1/2 6 3U − 4J
Table 3.2: Eigenstates and eigenvalues of the t2g Hamiltonian (3.13) in the atomic limit. The boxed
numbers denote the ground state degeneracies for J > 0.
3.3.1
Ising form of Kanamori model
We pointed out that from the form of the multi-orbital Hubbard model (3.13) the S U(2) symmetry of the Hund’s spin-spin coupling −2JS2is evident. However, it is quite common, for the practical reasons, to retain just the z-component of spinS. This means that, starting from the Hamiltonian (3.10), one needs to keep just the first three density-density terms, omitting the off-diagonal ones, spin-flip and pair-exchange. In this way we obtain the "Ising type" of Hund’s coupling, −2JS2z, which obviously does not preserve anymore the S U(2) symmetry. This approximation turns out to be extremely useful both for numerical solutions (e.g. Continuous-Time Quantum
